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In the realm of rocket motion, the velocity of a rocket over time is a crucial aspect that dictates its flight path and overall trajectory. For our test rocket, this velocity is described by the function \( v(t) = At + Bt^2 \). Here, \( A \) and \( B \) are constants that we need to determine. Understanding this function is essential for predicting how the rocket’s speed changes with time. To break it down:

• \( A \) represents an initial rate of change of velocity, immediately after ignition.
• \( B \) exemplifies the influence of acceleration changes over time, as \( Bt^2 \) contributes a quadratic element to the velocity curve.

Given the initial conditions, where the velocity at \( t = 1 \) second is 2.00 m/s, we use this equation to solve for \( B \). Substituting into the velocity equation, we find \( A = 1.50 \text{ m/s}^2 \) (from the initial acceleration) and \( B = 0.50 \text{ m/s}^2 \) (from the condition at \( t = 1 \)). Understanding the velocity function helps us predict how fast the rocket will go and how this speed changes as time progresses.

Acceleration is the rate at which the velocity of an object changes with time. In the context of our rocket, this is calculated by differentiating the velocity function. From the equation \( v(t) = At + Bt^2 \), the acceleration function is given as \( a(t) = \frac{dv(t)}{dt} = A + 2Bt \).Key points to remember:

• The constant \( A \) represents the initial acceleration at the moment of ignition. At \( t=0 \), thus \( a(0) = A = 1.50 \text{ m/s}^2 \).
• The term \( 2Bt \) indicates that acceleration increases linearly over time, contributing to the rocket's evolving motion.

To specifically find the acceleration at 4 seconds after ignition, we substitute \( t = 4 \) into the function, resulting in \( a(4) = 1.50 + 2(0.50)(4) = 5.50 \text{ m/s}^2\). Analyzing the acceleration helps us understand how quickly the rocket's velocity is increasing and is an indicator of the rocket's performance over its flight.

Thrust force is a crucial component in rocket physics. It is the force that propels the rocket upward, working against gravity and other forces. Calculating the thrust involves considering both the forces acting on the rocket and its mass.To compute thrust at a specific time, say 4 seconds after ignition:

• The net force acting on the rocket is given by the difference between the thrust force \( F_{thrust} \) and the gravitational pull \( F_{gravity} \).
• We use the formula \( F_{net} = ma = F_{thrust} - F_{gravity} \).

The gravitational force is calculated as \( F_{gravity} = mg = 2540 \times 9.81 \). Thus, the thrust force at 4 seconds is determined by solving the equation: \[ F_{thrust} = F_{gravity} + ma = 2540 \times (9.81 + 5.50) \].Initial thrust, when \( t=0 \), involves the same formula with initial acceleration: \[ F_{thrust} = 2540 \times (9.81 + 1.50) \].The thrust force is not only crucial for calculating the lift-off power of the rocket but also for ensuring that it remains on course, overcoming the gravitational pull, and achieving the desired altitude.